Theorem ssun1 3100

Description: Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ssun1	|- A C_ (A u. B)

Proof of Theorem ssun1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 632	. . 3 |- (x e. A -> (x e. A \/ x e. B))
2		elun 3078	. . 3 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
3	1, 2	sylibr 137	. 2 |- (x e. A -> x e. (A u. B))
4	3	ssriv 2943	1 |- A C_ (A u. B)

Syntax hints:   \/ wo 628   e. wcel 1390   u. cun 2909   C_ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925

This theorem is referenced by:  ssun2  3101  ssun3  3102  elun1  3104  inabs  3162  reuun1  3213  un00  3257  undifabs  3294  undifss  3297  snsspr1  3503  snsstp1  3505  snsstp2  3506  prsstp12  3508  sssucid  4118  unexb  4143  dmexg  4539  fvun1  5182  dftpos2  5817  tpostpos2  5821  ressxr  6866  nnssnn0  7960  un0addcl  7991  un0mulcl  7992  bdunexb  9375